Group #15766027 — subgroups

Subgroups — #15766027

Order 3328.

402 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123223	C₂	1	1	yes	maximal, characteristic, complemented
2	#13113396	C₂	1	2	yes	maximal
3	#13112696	C₂	1	2	yes	maximal, complemented
4	#13112669	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1962	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#532	C₂²	1	1	yes	characteristic
7	C₁₃×#423	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#104	—	2	2	no	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	#11718597	C₂²	1	2	yes	—
11	#11718594	—	2	4	no	—
12	#11718589	—	2	2	no	—
13	#11718586	C₂²	1	2	yes	—
14	#11718129	—	2	4	no	—
15	C₂×C₁₃×#41	—	2	1	no	—
16	C₂×C₁₃×M₁₆	—	2	1	no	—
17	C₁₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
18	C₁₃×#127	Q₈	1	1	yes	characteristic
19	C₁₃×#124	—	2	1	no	—
20	C₁₃×#121	—	2	2	no	—
21	C₄×D₄×C₁₃	—	2	2	no	—
22	C₂×C₄²×C₁₃	—	2	2	no	—
23	C₂×C₄²×C₁₃	D₄	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₃	—	2	2	no	—
26	C₂×#6101	—	2	4	no	—
27	C₁₃×C₄⋊C₈	—	2	2	no	—
28	C₁₃×#104	—	4	4	no	—
29	#88071	—	2	4	no	—
30	#88071	—	2	2	no	—
31	#88071	D₄	1	4	yes	—
32	C₂²⋊#1055	—	2	2	no	—
33	C₄⋊#1055	—	2	2	no	—
34	C₄×#1055	—	2	2	no	—
35	#10366	—	13	1	no	maximal, Sylow
36	C₁₃×#41	—	4	2	no	—
37	C₁₃×#41	—	4	2	no	—
38	C₂×Q₈×C₁₃	—	2	1	no	—
39	C₂×D₄×C₁₃	—	2	1	no	—
40	C₂×D₄×C₁₃	—	2	2	no	—
41	C₂×D₄×C₁₃	—	4	1	no	—
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	—	4	1	no	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	—	2	4	no	—
47	C₁₃×M₁₆	—	4	2	no	—
48	C₂×C₈×C₁₃	—	4	1	no	—
49	C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₁₃×C₂²⋊C₄	—	4	2	no	—
51	C₄²×C₁₃	—	2	2	no	—
52	C₄²×C₁₃	—	2	2	no	—
53	C₄²×C₁₃	—	4	4	no	—
54	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₁₃	#41	1	1	yes	characteristic
56	C₄²×C₁₃	#41	1	2	yes	—
57	C₄⋊Dic₁₃	—	4	2	no	—
58	C₄⋊Dic₁₃	—	4	2	no	—
59	#6101	—	4	8	no	—
60	C₂×#1055	—	4	2	no	—
61	#2864	—	13	1	no	—
62	#1962	D₁₃	1	1	yes	characteristic
63	#1517	—	13	2	no	—
64	#1286	—	13	2	no	—
65	#1276	—	13	1	no	—
66	C₂³×C₁₃	—	4	1	no	—
67	C₂³×C₁₃	#120	1	1	yes	characteristic
68	Q₈×C₁₃	—	4	2	no	—
69	D₄×C₁₃	—	4	2	no	—
70	D₄×C₁₃	—	8	2	no	—
71	D₄×C₁₃	—	8	2	no	—
72	C₂×C₄×C₁₃	—	2	4	no	—
73	C₂×C₄×C₁₃	—	4	2	no	—
74	C₂×C₄×C₁₃	—	4	2	no	—
75	C₂×C₄×C₁₃	—	4	2	no	—
76	C₂×C₄×C₁₃	—	4	1	no	—
77	C₂×C₄×C₁₃	—	8	1	no	—
78	C₂×C₄×C₁₃	—	8	1	no	—
79	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#124	1	2	yes	—
82	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#135	1	2	yes	—
84	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×Dic₁₃	—	4	8	no	—
86	C₂×Dic₁₃	—	4	4	no	—
87	C₈×C₁₃	—	8	1	no	—
88	#1055	—	8	2	no	—
89	#532	C₂×D₁₃	1	1	yes	characteristic
90	#431	—	13	2	no	—
91	#423	C₂×D₁₃	1	1	yes	characteristic
92	C₂×#104	—	2	2	no	—
93	#389	—	26	2	no	—
94	#387	—	26	4	no	—
95	#382	—	13	2	no	—
96	C₄³	C₂×D₁₃	1	1	yes	characteristic
97	#327	—	26	4	no	—
98	C₂²×C₁₃	—	2	2	no	—
99	C₂²×C₁₃	—	8	1	no	—
100	C₂²×C₁₃	—	8	2	no	—
101	C₂²×C₁₃	#393	1	1	yes	characteristic
102	C₂²×C₁₃	#449	1	2	yes	—
103	C₄×C₁₃	—	2	2	no	—
104	C₄×C₁₃	—	2	2	no	—
105	C₄×C₁₃	—	4	4	no	—
106	C₄×C₁₃	—	8	1	no	—
107	C₄×C₁₃	#478	1	2	yes	—
108	C₄×C₁₃	#488	1	1	yes	characteristic
109	C₄×C₁₃	#490	1	1	yes	characteristic
110	Dic₁₃	—	8	4	no	—
111	C₂×#41	—	2	1	no	—
112	C₂×M₁₆	—	2	1	no	—
113	C₄⋊Q₈	Dic₂₆	1	1	yes	characteristic
114	#127	Dic₂₆	1	1	yes	characteristic
115	#124	—	2	1	no	—
116	#121	—	2	2	no	—
117	C₄×D₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	26	4	no	—
119	C₂×C₄⋊C₄	—	26	2	no	—
120	C₂×C₄²	—	2	2	no	—
121	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
122	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	26	2	no	—
125	#104	—	4	4	no	—
126	C₂²⋊C₈	—	26	2	no	—
127	C₄×C₈	—	26	2	no	—
128	#95	—	13	4	no	—
129	#95	—	26	4	no	—
130	#95	—	26	2	no	—
131	C₂×C₁₃	—	2	2	no	—
132	C₂×C₁₃	—	16	1	no	—
133	C₂×C₁₃	#2025	1	1	yes	characteristic
134	C₂×C₁₃	#2031	1	2	yes	—
135	#41	—	4	2	no	—
136	#41	—	4	2	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	2	1	no	—
140	C₂×D₄	—	4	1	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	26	4	no	—
144	C₂²×C₄	D₄×D₁₃	1	2	yes	—
145	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
146	M₁₆	—	4	2	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	52	2	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	52	8	no	—
151	C₄⋊C₄	—	52	2	no	—
152	C₄⋊C₄	—	52	2	no	—
153	C₂²⋊C₄	—	4	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	4	4	no	—
157	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
158	C₄²	#6127	1	1	yes	characteristic
159	C₄²	#6129	1	2	yes	—
160	C₁₃	#10366	1	1	yes	characteristic, complemented, Sylow
161	C₂³	—	4	1	no	—
162	C₂³	#88191	1	1	yes	characteristic
163	Q₈	—	4	2	no	—
164	D₄	—	4	2	no	—
165	D₄	—	8	2	no	—
166	D₄	—	8	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	8	1	no	—
173	C₂×C₄	—	8	1	no	—
174	C₂×C₄	—	52	8	no	—
175	C₂×C₄	—	52	4	no	—
176	C₂×C₄	#88132	1	2	yes	—
177	C₂×C₄	#88139	1	2	yes	—
178	C₂×C₄	#88178	1	1	yes	characteristic
179	C₂×C₄	#88186	1	2	yes	—
180	C₂×C₄	#88190	1	1	yes	characteristic
181	C₂×C₄	#88204	1	2	yes	—
182	C₈	—	8	1	no	—
183	C₈	—	104	2	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	8	1	no	—
186	C₂²	—	8	2	no	—
187	C₂²	#11718608	1	1	yes	characteristic
188	C₂²	#11718698	1	2	yes	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	4	no	—
192	C₄	—	8	1	no	—
193	C₄	—	104	4	no	—
194	C₄	#11718670	1	2	yes	—
195	C₄	#11718717	1	1	yes	characteristic
196	C₄	#11718732	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	—	16	1	no	—
199	C₂	#13115982	1	1	yes	characteristic
200	C₂	#13116008	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13123223

C₂

yes

maximal, characteristic, complemented

#13113396

C₂

yes

maximal

#13112696

C₂

yes

maximal, complemented

#13112669

C₂

yes

maximal, characteristic, complemented

C₁₃×#1962

C₂

yes

maximal, characteristic

C₁₃×#532

C₂²

yes

characteristic

C₁₃×#423

C₂²

yes

characteristic

C₂×C₁₃×#104

—

C₄³×C₁₃

C₂²

yes

characteristic

#11718597

C₂²

yes

—

#11718594

—

#11718589

—

#11718586

C₂²

yes

—

#11718129

—

C₂×C₁₃×#41

—

C₂×C₁₃×M₁₆

—

C₁₃×C₄⋊Q₈

Q₈

yes

characteristic

C₁₃×#127

Q₈

yes

characteristic

C₁₃×#124

—

C₁₃×#121

—

C₄×D₄×C₁₃

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×#6101

—

C₁₃×C₄⋊C₈

—

C₁₃×#104

—

#88071

—

#88071

—

#88071

D₄

yes

—

C₂²⋊#1055

—

—

—

—

C₁₃×#41

—

C₁₃×#41

—

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×Q₈

yes

C₄²×C₁₃

yes

C₄²×C₁₃

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

D₁₃

yes

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

#120

yes

characteristic

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

#124

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

#1055

—

#532

C₂×D₁₃

yes

characteristic

#431

—

#423

C₂×D₁₃

yes

—

—

—

—

C₄³

C₂×D₁₃

yes

characteristic

#327

—

C₂²×C₁₃

—

C₂²×C₁₃

—

100

C₂²×C₁₃

—

101

C₂²×C₁₃

#393

yes

characteristic

102

C₂²×C₁₃

#449

yes

—

103

C₄×C₁₃

—

104

C₄×C₁₃

—

105

C₄×C₁₃

—

106

C₄×C₁₃

—

107

C₄×C₁₃

#478

yes

—

108

C₄×C₁₃

#488

yes

characteristic

109

C₄×C₁₃

#490

yes

characteristic

110

Dic₁₃

—

111

C₂×#41

—

112

C₂×M₁₆

—

113

C₄⋊Q₈

Dic₂₆

yes

characteristic

114

#127

Dic₂₆

yes

characteristic

115

#124

—

116

#121

—

117

C₄×D₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄²

—

121

C₂×C₄²

C₂²⋊D₁₃

yes

—

122

C₂×C₄²

C₂²×D₁₃

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

#104

—

126

C₂²⋊C₈

—

127

C₄×C₈

—

128

#95

—

129

#95

—

130

#95

—

131

C₂×C₁₃

—

132

C₂×C₁₃

—

133

C₂×C₁₃

#2025

yes

characteristic

134

C₂×C₁₃

#2031

yes

—

135

#41

—

136

#41

—

137

C₂×Q₈

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

D₄×D₁₃

yes

—

145

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

146

M₁₆

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₂²⋊C₄

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

C₂×Dic₂₆

yes

characteristic

158

C₄²

#6127

yes

characteristic

159

C₄²

#6129

yes

—

160

C₁₃

#10366

yes

characteristic, complemented, Sylow

161

C₂³

—

162

C₂³

#88191

yes

characteristic

163

Q₈

—

164

D₄

—

165

D₄

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

#88132

yes

—

177

C₂×C₄

#88139

yes

—

178

C₂×C₄

#88178

yes

characteristic

179

C₂×C₄

#88186

yes

—

180

C₂×C₄

#88190

yes

characteristic

181

C₂×C₄

#88204

yes

—

182

C₈

—

183

C₈

—

104

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

#11718608

yes

characteristic

188

C₂²

#11718698

yes

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

—

104

—

194

C₄

#11718670

yes

—

195

C₄

#11718717

yes

characteristic

196

C₄

#11718732

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#13115982

yes

characteristic

200

C₂

#13116008

yes

—

	subgroup	G/H
center	C₂²	#11718608
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1962	C₂
socle	C₂²×C₁₃	#393
prime radical	C₁₃×#532	C₂²
hypercenter	#1962	D₁₃
hypocenter	C₁₃	#10366
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1962

subgroup

G/H

center

C₂²

#11718608

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1962

C₂

socle

C₂²×C₁₃

#393

prime radical

C₁₃×#532

C₂²

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1962

N	H ≅ G/N	type
1. #13123223	C₂	semidirect
3. #13112696	C₂	semidirect
4. #13112669	C₂	semidirect
160. C₁₃	#10366	semidirect

H ≅ G/N

type

1. #13123223

C₂

semidirect

3. #13112696

C₂

semidirect

4. #13112669

C₂

semidirect

160. C₁₃

#10366

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13123223	2
#13113396	2
#13112696	2
#13112669	2
C₁₃×#1962	2
#10366	13

subgroup

index

C₁₃×#1962

Subgroups — #15766027

Conjugacy classes of subgroups

Complements

Maximal subgroups